Are there any other board games that have similar game mechanics to Risk?
I generally think of Axis and Allies as being like advanced Risk. Also there are all of the different Risk clones, 2210, LoTR, Star Wars, Godstorm and others which introduce slight variants in maps and rules. Of those I'd recommend 2210, but if you can handle the complexity, which isn't bad in the grand scheme, Axis and Allies is the way to go.
Diplomacy is a similar game in that its aim is to conquer other players land. However it removes the dice and the element of chance, replacing it with face to face negotiation (making it a much better game IMHO).
Shogun is another similar game. The combat is different in this game (you are not using dice at all) but gathering resources and conquering other provinces (which act like countries in Risk) has the same basic principles.
History of the World is played much like Risk. There are 7 epochs. During each epoch a player takes control of a different empire from the past. They expand and conquer as much as they can (this part plays out very similar to Risk). Next epoch they play a new vibrant empire.
Scoring is based on area control at the end of each player's turn.
Last year a new edition was released called, A Brief History of the World which is supposed to play to completion quicker. I haven't yet tried it.
There is also Dust, which is set in an alternate World War II. There you have mechs and tanks, submarines and destroyers, bombers and fighters. New additions also include character-based powers, fortified positions and capitals. Unlike vanilla RISK, it uses a VP track to determine victory.
One of the interesting part of the combat is that tanks soak damage before mechs, fighters soak damage before bombers and destroyers soak damage before submarines (If I recall correctly). Mechs, Bombers and Submarines provide more offense in the game
The official site for the game is here.
How about Twilight Imperium. We always play and when I explain to someone who is not into the scene what it is I always end up saying:"Ok, do you know Risk? Well it's like that but more complicated".
But I guess there are loads of game that then qualify as advanced Risk :).
Ozymandia is simultaneous Risk, simplified to its essence.
I used to play Samurai Swords (also known as Shogun) as a kid. It's a lot like Risk, with several unit types, more complex battles, and more territories. It takes forever though.
I'm not sure if this is what you're looking for, but there are various Risk variants that use exactly the same rules and mechanics, but on a different board. I've got the Lord of the Rings and Star Wars variants, and there are others. As far as I can tell (from the LOTR and Star Wars verions) they normally/often have some extra (and optional) themed rule and unit variations, which, combined with the new map topology, can keep it interesting if you're bored of classic Risk.