I had to take several seminars on communication when I worked for a local government agency, and this topic was discussed at all of them. We were taught four main strategies.
1. Mirroring. This simply refers to watching what the other person is doing and acting in a similar manner. If they use a soft tone of voice and speak slowly, try and match your speech to their volume and pace. If they use gestures, such as nodding the head to signify understanding, feel free to also nod in understanding as they speak to you. If their posture is casual, such as standing with their hands in their pockets, you can relax your posture a bit was well; if their posture is very straight and stiff, try to mirror theirs. The goal of mirroring is to set the other person at ease. Gradually, as you become more familiar with one another, your own, personal or cultural tendencies will tend to emerge more, but mirroring at the beginning is a good way for you both to establish a safe comfort zone.
2. Body posture. Generally, it is best to practice "open" body posture. Keep your head up, rather than tucked down into your chin, and don't cross your arms or your legs. Try and keep your facial expression as open as possible as well by smiling and maintaining polite eye contact. Opening your body posture is a way of communicating openness and your willingness to communicate.
3. Verbal expression. As noted above in item #1, you should try and match the tone and speed of your speech to theirs. You should also do this with vocabulary. If you are communicating across language barriers, keep it simple. Don't say "enormous" when "big" will do just fine. If you both speak the same language fluently, still make an effort to match your vocabulary to theirs. Don't make assumptions based on appearances, such as talking down to someone because they look like they may not have the same educational background as you do. (Listening to verbal cues, such as how they pronounce words and the fluency with which they express themselves will usually help you out in this department.) Try to avoid slang and regionalisms, as these can cause a lot of confusion. Most importantly, if you don't understand something, politely ask for clarification.
4. Research. This is invaluable, especially if you know in advance that you are going to be dealing with someone from a culture which is considerably different from your own. Going to Japan as an exchange student? Read up on Japanese culture and social customs. People flying in from the Rome office for a big meeting? Read up on Italian culture so you have some idea of what to expect.
They also taught us that in dealing with people from other countries and cultural backgrounds, there are several areas where it is easy to get into trouble, so be careful about these:
--"Personal space". In the US, "personal space", meaning how close you physically are from someone when speaking to them, is usually 3-4 feet. The tricky thing is that "personal space" varies widely from culture to culture. In some cultures, people maintain a greater distance when communicating, and in others, they get so close that many Americans feel like they are being invaded. It's very hard not to take a step back when someone steps close to you, or not to step closer to them if they step away. Try your best to let them set the boundary, and then try to keep that distance constant. However, if they are so close to you that you are uncomfortable, and cannot concentrate on the conversation, it is acceptable to take a small step back after a few moments.
--Physical contact: Again, this one varies greatly from place to place. Shaking hands can really be a problem. If someone offers you their hand, shake it. If you put yours out first, and they don't make a move to take it, return you hand to your side, don't just grab theirs and shake away.