At least for the Xbox 360, this is largely attributed to the Achievement system, which was meant to publicly acknowledge gamers' abilities and accomplishments. If cheat codes were prevalent, then finishing a game wouldn't be quite as impressive. Of course, many modern movie tie-ins and budget games do offer codes to input; but for the most part things like “God Mode” are left as unlockables or completely left out.
For the most part, I see the elimination of cheating as a good thing. It supports the important maxim of “play the game as it was meant to be played.” As the video-game industry climbs in public approval and as a story telling medium, removing cheat codes was an important step to becoming mature. At least stereotypically, cheats were often associated with children or gamers incapable of properly playing video-games. It removed difficulty for the sake of fan-service, and was an invitation from the developer to completely ignore all of their hard work. Obviously “big head mode” was not a plague ruining the industry, and Mortal Kombat's kodes were never hurting anyone. There is a fine line between the codes that have provided fond memories and nostalgia for years to come, and the ones which simply ruined a game.
Nowadays the trend is to be rewarded with bonuses, as opposed to immediately being able to acquire them. Dead Rising for instance, offers extremely powerful weaponry to those willing to kill thousands and thousand of zombies. Not quite as simple as inputting the Konami code, this means of unlocking a modern “cheat” is more fair and worthwhile. Even still, anything as game-breaking as unlimited ammo or health is mostly taboo as of today.
Even the term “cheating” has changed greatly with the dawn of casual online multiplayer and versus games. For console first-person shooters, a cheater is most typically an in-game hacker or abuser to in-game glitches. Games have been modified using outside tools for ages, but with versus gametypes the potential for ruining others' fun is far greater. There will always be cheaters in gaming, even if the terminology is drastically different than we mean today.
Quite honestly I do not miss the majority of cheat codes. The ones which strived for humor are usually dated, or simply not worth your time. While most codes were easily added to the game by modifying certain elements of the game's code, it really can be seen as a waste of time. The days of cheating profusely might be fondly remembered by some, but to me they either ruined a game by reducing its difficulty or by mocking itself. At the same time, it really depends on which cheats you're talking about. Grand Theft Auto's codes got me out of hundreds of near-deaths and arrests, and even Mortal Kombat 9 pulled off some hilarious extra goodies.