Note that despite how the sub-headings might make it appear, this answer is directly opposite to most of the other answers.
Many Christians do abstain from pork and other unclean meat.
Seventh Day Adventists are perhaps the most famous, but they are far from alone in this belief.
There is plentiful literature on the subject, such as What Does the Bible Teach About Clean and Unclean Meats?, from a non-SDA church.
The other answers to this question give the standard list of proof texts on this subject, but it is mostly rationalization, using eisegesis to justify eating pork.
Initially, the prohibition against pork was part of the Law given to Moses [so it's binding only on Jews]
The distinction between clean and unclean meat existed long before Moses.
Notice God's instruction on what to take aboard the ark:
You shall take with you seven each of every clean animal, a male and his female; two each of animals that are unclean, a male and his female
Noah was the ancestor of all mankind, so if he had to distinguish between clean and unclean meat, it's hardly only a Jewish thing.
No more than murder and theft, which were also "given to Moses".
Jesus declared all foods clean
The Pharisees had accused his disciples of eating without first ceremonially washing their hands.
Jesus explained that they were hypocrites because a tiny bit of dirt on food would simply pass through and end up purged from the body into the toilet, while figurative dirt (what we listen to) can enter the heart and stay there affecting how we behave.
“Are you so dull?” he asked. “Don’t you see that nothing that enters a person from the outside can defile them?
For it doesn’t go into their heart but into their stomach, and then out of the body.” (In saying this, Jesus declared all foods clean.)
— Mark 7:18,19 (NIV)
Notice that the final statement is in parentheses. That's because the statement doesn't appear in the original Greek manuscripts; it has been added by the translators for "clarification".
More literal translations, such as the KJV, say:
And he saith unto them, Are ye so without understanding also? Do ye not perceive, that whatsoever thing from without entereth into the man, it cannot defile him;
Because it entereth not into his heart, but into the belly, and goeth out into the draught, purging all meats?
This is Shakespearean English, where "draught" means "latrine" and "meats" means "food".
If Jesus had declared all food clean, or even if the Pharisees only thought that was what he said, they would have reacted far far more for that violation of the law than for minor things like hand washing.
Surely such an incident would have been recorded by all four Gospels, but none of them even mention it.
That incident didn't happen, and that's because he didn't say it.
Matthew does record the same interaction with the Pharisees, and gives the same conclusion:
“Not what goes into the mouth defiles a man; but what comes out of the mouth, this defiles a man.”
So Jesus said, “Are you also still without understanding?
Do you not yet understand that whatever enters the mouth goes into the stomach and is eliminated?
But those things which proceed out of the mouth come from the heart, and they defile a man.
For out of the heart proceed evil thoughts, murders, adulteries, fornications, thefts, false witness, blasphemies.
These are the things which defile a man, but to eat with unwashed hands does not defile a man.” — Matthew 15:11,16–20
Had Jesus actually declared all meat to be clean, surely Matthew would have recorded it here.
But he didn't.
God told Peter that all meat was clean
And a voice came to him, “Rise, Peter; kill and eat.”
But Peter said, “Not so, Lord! For I have never eaten anything common or unclean.”
And a voice spoke to him again the second time, “What God has cleansed you must not call common.”
— Acts 10:13–15
Peter didn't understand what it meant: "Now while Peter wondered within himself what this vision which he had seen meant, …".
But later he did realize its meaning:
You know how unlawful it is for a Jewish man to keep company with or go to one of another nation. But God has shown me that I should not call any man common or unclean.
— Acts 10:28
Peter understood what the vision meant, but many people today choose to think it means something completely different.
Now notice that Peter said "I have never eaten anything common or unclean.".
If Jesus had "declared all foods clean", why would Peter still be refraining from eating it many years later?
And why would he, in the vision, have refused to eat it when told to?
The only explanation that makes sense, without massive rationalization, is that Jesus never said such a thing.
Peter said that nothing is unclean
People tend to quote the last sentence in this passage without considering the rest:
Receive one who is weak in the faith, but not to disputes over doubtful things.
For one believes he may eat all things, but he who is weak eats only vegetables.
Let not him who eats despise him who does not eat, and let not him who does not eat judge him who eats; for God has received him.
Therefore let us not judge one another anymore, but rather resolve this, not to put a stumbling block or a cause to fall in our brother’s way.
I know and am convinced by the Lord Jesus that there is nothing unclean of itself; but to him who considers anything to be unclean, to him it is unclean.
— Romans 14:1–3,13–14
Peter was talking about new converts, many of whom were vegetarians.
Rather than concentrating on teaching the fundamentals of Christianity, many people were more concerned with far less important things, such as getting the converts to eat meat.
This was distressing to the converts and they were in danger of leaving the church because of it.
Peter wanted his fellow Christians to know that the vegetarians will eventually learn to understand that it's okay to eat meat, but in the meantime they should be left alone so that they can learn the important parts of Christianity first.
In context, Peter's statement is referring to converting vegetarians that don't want to eat clean meat, such as lamb or fish, along with the other people, and the importance of not making them feel uncomfortable about it.
So let no one judge you in food or in drink
So let no one judge you in food or in drink, or regarding a festival or a new moon or sabbaths — Colossians 2:16
Again, people that already believe that pork is edible use this verse to support their position.
But in context, it actually means quite the opposite.
It is the Christians that are being criticised by vegetarians and ascetics for eating meat, and Paul is telling them not to let those other, non-Christians, judge them against what they consider their own superior morality.
I give a much fuller explanation of this reference in
exegesis - Keeping the Sabbath and Colossians 2:16 - Christianity Stack Exchange
It's only Old Testament
Isaiah 66 is considered a messianic prophecy, when Christ returns during the Day of the Lord to rule during the Millennium:
“Behold, I will extend peace to her like a river,
And the glory of the Gentiles like a flowing stream.
For behold, the LORD will come with fire
And with His chariots, like a whirlwind,
To render His anger with fury,
And His rebuke with flames of fire.
For by fire and by His sword
The LORD will judge all flesh;
And the slain of the LORD shall be many.
Those who sanctify themselves and purify themselves,
To go to the gardens
After an idol in the midst,
Eating swine’s flesh and the abomination and the mouse,
Shall be consumed together,” says the LORD.
— Isaiah 66:12,15–17
This is still some time in our future, but notice that "eating swine's flesh" is one of the things that people are condemned for.