The influence of American action movies in Japan has been huge. Shortly after the success of all the 1980s Hollywood action flicks with dudes like Schwarzenegger, Stallone and Chuck Norris, Japanese gaming created an entire library of different action titles with one common feature: Kill everything on the screen. Oh, and blow up everything, too.
Capcom's Commando, a vertically-scrolling shoot'em up that was ported to basically everything that was available in 1985, let us control a badass dude called Super Joe.
SNK's Ikari Warriors has a reputation. Let's just say gameplay is...nothing like the Metal Slug series.
Bionic Commando is an action platformer in which you had to infiltrate an enemy base to prevent a deadly missile launch. The game was advertised in North America improperly as the sequel to Commando, but that was actually another game called Mercs.
Rambo: First Blood Part II was an action shooter based on the second Rambo movie, released by Sega in 1986. An interesting fact that in the Japanese version, you had to control Buddhist monks, and it had no connection at all with Rambo (the game came out without the Rambo license).
Data East's Heavy Barrel was an arcade action game similar to Ikari Warriors (but it's a lot better). You had to prevent - AGAIN - the launch of nuclear missiles.
The Metal Slug series do not need an introduction. These games are still being produced by SNK, and it is still not that hard to find an arcade in Japan with a Metal Slug machine.
Jackal (know as Top Gunner in North America, released in 1988) was one of Konami's entry in the run-and-gun genre.
You can argue which Contra game is the best one, but the superb gameplay and the special, 3D isometric stages made all the early titles fantastic.
Devastators' (Garuka in Japan, released by Konami) had similar gameplay to Contra's 3D stages. The background is slowly moving towards us, while we're busy shooting down everyone.
Vanquish - Platinum Games' fast paced (or in this case, super fast paced) third person action title - is the best example that these Japanese games with a badass American protagonist aren't going away any time soon.
sources: MamePlayer, Offhand Games, skorch82, BridgetPlays, Rinatabdr, EightbitHD, retrojuegoschile