The LDS concept of preaching to the spirits of the dead derives from a couple passages in 1 Peter:
18 For Christ also hath once suffered for sins, the just for the unjust, that he might bring us to God, being put to death in the flesh, but quickened by the Spirit:
19 By which also he went and preached unto the spirits in prison;
20 Which sometime were disobedient, when once the longsuffering of God waited in the days of Noah, while the ark was a preparing, wherein few, that is, eight souls were saved by water.
1 Peter 3: 18-20
6 For for this cause was the gospel preached also to them that are dead, that they might be judged according to men in the flesh, but live according to God in the spirit.
1 Peter 4: 6
which were later elaborated on in a vision to the prophet Joseph F. Smith (not Joseph Smith, who restored the church, but he was related to him) which is preserved in the book of Doctrine and Covenants:
11 As I pondered over these things which are written, the eyes of my
understanding were opened, and the Spirit of the Lord rested upon me,
and I saw the hosts of the dead, both small and great.
12 And there were gathered together in one place an innumerable
company of the spirits of the just, who had been faithful in the
testimony of Jesus while they lived in mortality;
13 And who had offered sacrifice in the similitude of the great
sacrifice of the Son of God, and had suffered tribulation in their
14 All these had departed the mortal life, firm in the hope of a
glorious resurrection, through the grace of God the Father and his
Only Begotten Son, Jesus Christ.
15 I beheld that they were filled with joy and gladness, and were
rejoicing together because the day of their deliverance was at hand.
16 They were assembled awaiting the advent of the Son of God into the
spirit world, to declare their redemption from the bands of death.
17 Their sleeping dust was to be restored unto its perfect frame, bone
to his bone, and the sinews and the flesh upon them, the spirit and
the body to be united never again to be divided, that they might
receive a fulness of joy.
18 While this vast multitude waited and conversed, rejoicing in the
hour of their deliverance from the chains of death, the Son of God
appeared, declaring liberty to the captives who had been faithful;
19 And there he preached to them the everlasting gospel, the doctrine
of the resurrection and the redemption of mankind from the fall, and
from individual sins on conditions of repentance.
20 But unto the wicked he did not go, and among the ungodly and the
unrepentant who had defiled themselves while in the flesh, his voice
was not raised;
21 Neither did the rebellious who rejected the testimonies and the
warnings of the ancient prophets behold his presence, nor look upon
22 Where these were, darkness reigned, but among the righteous there
23 And the saints rejoiced in their redemption, and bowed the knee and
acknowledged the Son of God as their Redeemer and Deliverer from death
and the chains of hell.
24 Their countenances shone, and the radiance from the presence of the
Lord rested upon them, and they sang praises unto his holy name.
25 I marveled, for I understood that the Savior spent about three
years in his ministry among the Jews and those of the house of Israel,
endeavoring to teach them the everlasting gospel and call them unto
26 And yet, notwithstanding his mighty works, and miracles, and
proclamation of the truth, in great power and authority, there were
but few who hearkened to his voice, and rejoiced in his presence, and
received salvation at his hands.
27 But his ministry among those who were dead was limited to the brief
time intervening between the crucifixion and his resurrection;
28 And I wondered at the words of Peter—wherein he said that the Son
of God preached unto the spirits in prison, who sometime were
disobedient, when once the long-suffering of God waited in the days of
Noah—and how it was possible for him to preach to those spirits and
perform the necessary labor among them in so short a time.
29 And as I wondered, my eyes were opened, and my understanding
quickened, and I perceived that the Lord went not in person among the
wicked and the disobedient who had rejected the truth, to teach them;
30 But behold, from among the righteous, he organized his forces and
appointed messengers, clothed with power and authority, and
commissioned them to go forth and carry the light of the gospel to
them that were in darkness, even to all the spirits of men; and thus
was the gospel preached to the dead.
31 And the chosen messengers went forth to declare the acceptable day
of the Lord and proclaim liberty to the captives who were bound, even
unto all who would repent of their sins and receive the gospel.
32 Thus was the gospel preached to those who had died in their sins,
without a knowledge of the truth, or in transgression, having rejected
33 These were taught faith in God, repentance from sin, vicarious
baptism for the remission of sins, the gift of the Holy Ghost by the
laying on of hands,
34 And all other principles of the gospel that were necessary for them
to know in order to qualify themselves that they might be judged
according to men in the flesh, but live according to God in the
35 And so it was made known among the dead, both small and great, the
unrighteous as well as the faithful, that redemption had been wrought
through the sacrifice of the Son of God upon the cross.
36 Thus was it made known that our Redeemer spent his time during his
sojourn in the world of spirits, instructing and preparing the
faithful spirits of the prophets who had testified of him in the
37 That they might carry the message of redemption unto all the dead,
unto whom he could not go personally, because of their rebellion and
transgression, that they through the ministration of his servants
might also hear his words.
-- from D&C Section 138
This explains how Jesus's work during "Holy Saturday" was to prepare the righteous dead for resurrection and also to establish and organize the work of preaching the Gospel to the unrighteous dead, which work began after the resurrection and has continued ever since.
Latter-Day Saints hold that this doctrine shows the love and fairness of God and his plan: that, as you put it, "the dude in the jungle" will have the same opportunity to hear and accept the Gospel as those born under more fortunate circumstances had. However, this part of your understanding is not accurate:
Presumably there is no time limit set on the offer of salvation and therefore it is safe to assume that everyone will eventually end up responding positively to the gospel (either here on earth or after death in Hell).
Dead people are still people. They have their own free will, and are free to accept or reject the Gospel message when it is taught to them. There is also eventually a time limit: the end of the world. When the work of mortality is done and the final judgment arrives, the wicked will be called forth and resurrected, and judged according to their works, as described in the Scriptures. In fact, the Book of Mormon explicitly warns that we still remain fundamentally ourselves after death:
33 And now, as I said unto you before, as ye have had so many witnesses, therefore, I beseech of you that ye do not procrastinate the day of your repentance until the end; for after this day of life, which is given us to prepare for eternity, behold, if we do not improve our time while in this life, then cometh the night of darkness wherein there can be no labor performed.
34 Ye cannot say, when ye are brought to that awful crisis, that I will repent, that I will return to my God. Nay, ye cannot say this; for that same spirit which doth possess your bodies at the time that ye go out of this life, that same spirit will have power to possess your body in that eternal world.
35 For behold, if ye have procrastinated the day of your repentance even until death, behold, ye have become subjected to the spirit of the devil, and he doth seal you his; therefore, the Spirit of the Lord hath withdrawn from you, and hath no place in you, and the devil hath all power over you; and this is the final state of the wicked.
This warning tells us that it's a bad idea to rely on this doctrine to give ourselves free rein to knowingly reject the Gospel in this life and try to still reap the benefits of the Gospel afterwards.