If you are interested in "making games" consider Unity3D or Torque Game Engine.
Both of those now have ways to be embedded into a browser, but I don't think they work on Linux. On the other hand there are things like Wine which might help you with that.
I suggested these things because there are a lot of things that go along with the game engine like tools and exporters and those two have fairly complete feature sets. They have small pricetags but if you consider the amount of time you would have to spend making your own tools or messing around with the engines themselves to get things to work, they are dirt cheap. At least do yourself the favor of trying out those engines with their toolsets so you'll know what you're missing if you go to a less complete platform.
If you are talking about "making a 3D game", for the love of God don't go off trying to make your own engine and toolset. Most likely you will get bogged down working on boring, hard problems that others have solved many times before and have done a better job at it than you could do without years of work.
On the other hand, if you are talking about "making a game engine" or "making a game in 3D on Linux that can run in a browser" then you are only tangentially talking about making a game. Instead you are mostly talking about getting cross-platform 3D to work in a browser, which is mostly about making a game engine. In that case, go for something open source that you can hack to your heart's content and spend the time getting it the way you want. However, understand that you are solving a different problem than "making a 3D game" and it will be a much longer time before you get to "making a 3D game".