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That's only really a problem if you make the game both extremely loot-focused and extremely grind-heavy. Sure in a game like WoW, where every item is super awesomely cool, with hundreds of bonuses and so forth, and can only be obtained by a rare drop from some 25 man dungeon, decay would piss people off.
The problem is that you're making the sword itself a huge achievement. Getting it is an achievement, walking around town with your cool glowey-blue custom model sword is an achievement. And then assuming the player wont want to carry on getting new swords, and new items. In a game focused on diablo-style loot grabbing the player is going to be constantly getting new swords. If he gets a sword and it decays in a week of play-time it shouldn't matter, because by the end of the week (assuming it only tics down while he's in-game) he'll have a new sword and wont care.
There is more than highend drops when we consider achievment.
For example, if player character starts naked with several empty inventory slots, just act of filling one of them is achievment of its own kind.
Many players are collectors, collecting for example one of each sword design (or all stat permutations of one design).
Then you have people who are simply attached to one item (propably every mmo player has few memoralibia stashed)
As for having new sword by time you run out of it. Sure, but it would bug people. For first few swords, it could be even exciting to carry spare with you to have weapon when your current "runs out", but when it is X-th one? It just turns to managing shopping list. "my wednesdays groceries are, hmm, sword, chainmail shirt and left leather glove."
It does not even have to be grind as in thast you have spend hours killing same mob to get gold for it. Grind here is that you are going to do exactly same thing (buy new stuff on shopping list) over and over each week.