What is the opinion of the LDS Church concerning prayers offered to Saints, Mary, and Jesus Christ? Are they simply vain words? Are they in anyway acknowledged? Is it considered unrighteous to pray to anyone but God the Father?
First of all, God is the one who decides which prayers he answers, not us. I take it for granted that he hears all prayers, and it would be presumptuous to say prayers addressed to X go unheard.
Nevertheless, Christ taught that we ought to pray the Father, in his name. Let's just do it and not wonder whether he will answer other people's prayers or not. Generally, I feel Church doctrine is more about defining what is right, than to define what is wrong. Praying the Father in the name of Christ is the right thing to do, because Jesus said so.
Additionaly, I feel that God deserves the honor that we address him directly.
23 And in that day ye shall ask me nothing. Verily, verily, I say unto you, Whatsoever ye shall ask the Father in my name, he will give it you.
24 Hitherto have ye asked nothing in my name: ask, and ye shall receive, that your joy may be full.
25 These things have I spoken unto you in proverbs: but the time cometh, when I shall no more speak unto you in proverbs, but I shall shew you plainly of the Father.
26 At that day ye shall ask in my name: and I say not unto you, that I will pray the Father for you:
27 For the Father himself loveth you, because ye have loved me, and have believed that I came out from God.
And, while not directly related, I'd like to add:
And when they shall say unto you, Seek unto them that have familiar spirits, and unto wizards that peep, and that mutter: should not a people seek unto their God? for the living to the dead?
What I am trying to say is this: Do people pray to Saints because they think God himself won't hear/answer them? What are they trying to achieve by this?
Also slightly unrelated, this is part of what the Bible Dictionary has to say about prayer:
Christians are taught to pray in Christ’s name (John 14:13–14; 15:7, 16; 16:23–24). We pray in Christ’s name when our mind is the mind of Christ, and our wishes the wishes of Christ—when His words abide in us (John 15:7). We then ask for things it is possible for God to grant. Many prayers remain unanswered because they are not in Christ’s name at all; they in no way represent His mind but spring out of the selfishness of man’s heart.
Personally, I feel this is far more likely to be the cause of unanswered prayer than some formality coming from being taught one way or the other.