This is going to depend on a lot of things in the company culture and what country you are in. Most companies I have worked for would be pretty unlikely to approve 3 weeks time off for anything less than a personal illness or family emergency. Some would, but only for people who were returning to a home country for a visit. I personally have never worked anywhere that it would be automatically approved. So your feeling that they may object could be true. In Europe they have longer vacations, so this might be an easier sell there than the US.
What you need to do is look at the project schedule and determine when would be the best time for this and ask months ahead of time and, above all, do not buy any non-refundable tickets until you have approval. Depending on why you want to take this time off, it could help to tell them what your plans are (It could also hurt). But many places woudl be more likely to approve this for a one-time big event like a wedding and honeymoon than just for I want to sit around and do nothing for three weeks.
Often the end of the year holiday season is a good time for a long vacation as things are slower then and I have seen people take vacation the last two weeks of the year and the first week of the new year in order to take vacation time from two different leave years. However, it is best to get your request in early for that time as others will be asking to use their leave in that time frame as well. If I were asking for that long at the holidays, my request would have gone in in February or so. Since there is only one other person on your team, check with him or her about leave plans. You don't want to be asking for the same week he is planning to get married for instance.
Depending on what you are doing, consider if you can go somewhere and work remotely part of that time. I have known people who returned to their home country for a month and take two weeks leave and work remotely for 2 weeks. Of course if you are climbing Mt. Everest this might not be practical. But it is thought for an alternative if they don't want you gone that long.
I have a former coworker who loves to take long trips and he generally works for two years, quits, takes a trip for a couple of months and then looks for another job. You have to be pretty good at managing money and pretty good at what you do and confident you can find a new job quickly to take this approach though.