The easiest way would be to stack Chameleon spells and effects (via Alchemy, enchanted items, or a custom spell), until you have a total chameleon value of 100% or more. With 100% Chameleon, you can basically do whatever you want with impunity, as nobody can see you, and unlike invisibility, Chameleon does not break when you interact with the world around you.
See also: UESP Wiki on Chameleon in Morrowind
One of the main problems with Oblivion is that it doesn't have the same world depth that Morrowind had. Sure, it has dungeons littered about the landscape, and there's no shortage of adventuring, but there are not nearly as many factions. Morrowind had the 3 Houses, the Fighters/Mages/Thieves guild, the Morag Tong, the Imperial Cult/Legion, and I'm sure I'm missing some, all in addition to the main quest. Even then, some were mutually exclusive, making you really feel like you were in the world. If you got to the top of the Fighter's Guild, people reacted to you differently. They recognized you.
In Oblivion, you still have your 3 guilds, and you have the Dark Brotherhood. That's it. There are no other factions to join, and each one has somewhere on the order of 10-15 quests, including the little mini-quests at the beginning just to whet your appetite. It seems short. Not only that, but you can have one character complete all the factions. There are no stat requirements for advancement, just quest requirements. You can complete the Mage's Guild with little or no actual experience in magic, and they still call you the archmage. It destroys the immersiveness. Even so, many of the quests are very well-constructed and you feel accomplished for having completed them.
The combat system is vastly improved, and it is satisfying that any attack you make will hit and deal damage so long as the animation hits. One of the most annoying parts about Marksman in Morrowind was that even if you managed to hit with the arrow, it still had a chance to miss based on your Marksman skill.
Another major gripe is that most of Cyrodiil (the world for Oblivion) looks relatively the same. The area near Skingrad has a lot of colorful flowers and the area near Leyawiin looks vaguely like a marsh, but for the most part everything's green. The regions are not as diverse as they were in Morrowind, and you shouldn't expect to find any interesting fauna like mushroom trees about the landscape.
One of the main difficulties is the fact that the enemies scale to your level. If you don't plan your skills carefully and account for what enemies you'll be facing, you will quickly find yourself outmatched just during the normal gameplay progression. You need to have multiple sources of damage, or else you will come across battles that are either unwinnable or take an excess of time to win. For example, one swordsman I made could never kill skeletons, no matter how long I bashed away at them. All my swords would run out of durability before the skeleton died, so whenever I encountered them I either had to run or hide. This can be fun or not, depending on your play style.
Oblivion is a fun game, but it's not the same game. If you go in expecting to have Morrowind with enhanced gameplay, you'll be disappointed. If you go in just looking for a fun time, it is certainly a worthy title. It's worth playing through once or twice, and the Shivering Isles expansion packs is one of the most fun times I've ever had with gaming.
Best Answer
It appears to work similarly, though at a lower percentage.
Oblivion's Chameleon at 100% (emphasis mine):
So essentially in Oblivion, 100% Chameleon, as you stated in the question, makes you basically invisible, allowing you to get away with whatever you want.
A similar effect appears in Morrowind (emphasis mine):
So it seems that the same ability to move around undetected exists in Morrowind, though unlike in Oblivion, it occurs at 75% Chameleon (as opposed to 100%).