3 on 3 is still fairly new to the NHL, so I would imagine strategy is still well in development - teams have played tens of minutes at most of 3 on 3 outside of practice.
However, from the rest of the world, 3 on 3 is fairly common in lower level games, and some of the strategy there clearly makes it to the NHL game. Most of the differences arise from the larger spaces on the ice, and higher relevance of each individual player.
- Triangle formation on defense: This is the same as the 5 on 3 defense, really. One guy between each faceoff circle and the goal, and then the third 'point' around mid-ice in your zone. Cuts off passing lanes very effectively.
- Triangle formation on offense: Basically the same as the defense's triangle. One guy back, two guys forward. When a shot happens, whomever is on the opposite side needs to crash the net hard when that occurs to get the rebound.
- One defender and two forwards: This is somewhat of a result of the triangle formation, but it's a good general approach even if you use a different formation. One player needs to play closer to the blue line, behind any of the defending team's players, to avoid breakaways. Breakaways will happen constantly in 3 on 3, so that player needs to be fast - and maybe not a "D" man (someone like Jonathan Toews for example is a great choice - fast, hard skater, big presence to prevent breakaways).
- Forechecking can be even more important. When there are only three defenders, taking one of them out during the rush can be critical - even a few seconds of 3 on 2 can give you a score. It also can be critical to prevent breakaways.
- Frequent line changes are a must, and getting clean changes off. This keeps your players fresh, and avoids bad breakaways.
These are a combination of my limited experience watching 3 on 3 NHL hockey, and reading some articles on amateur 3 on 3 strategies. Some of those don't apply to the NHL (smaller ice, for example, is of course off the table), but there is interesting information there nonetheless. This article is the one I most used to source this.