The Deal
Wizards get: Forward Jared Dudley
Bucks get: Future second-round pick
Washington Wizards: B+

It's unfair to say that Jared Dudley is replacing Paul Pierce for the Wizards. While Dudley has had a fine career in his own right, he's not a 10-time All-Star or former champion like Pierce, and it's hard to see Dudley winning multiple playoff games with buzzer-beating jumpers. (Dudley did assist the game-winning layup for Milwaukee in Game 4 against Chicago, it's true.)
That said, with Pierce leaving to join his hometown L.A. Clippers, Washington was in need of a combo forward who could team with Otto Porter Jr. in the kinds of small units that were so effective for head coach Randy Wittman during the postseason. That kind of versatility has been a hallmark for Dudley, an undersized power forward at Boston College who worked to develop into a 3-and-D wing and has now returned to the 4 spot on a regular basis as he's aged.
While it was tricky to parse positions at times with the Bucks last season, based on lineup data from NBA.com/Stats I have Dudley playing 1,015 minutes at small forward and 608 at power forward, with the team outscoring its opponents with him at either position. Expect a similar split with the Wizards depending on Wittman's comfort using small lineups in the regular season, Porter's development and how Martell Webster rebounds from last year's back surgery.
The price for Dudley was right. From a salary perspective, he'll make an eminently reasonable $4.25 million this season after opting into the final year of his contract, which fits into a trade exception Washington has from dealing Andre Miller at the deadline. I project Dudley's value at $6 million. And it doesn't appear he cost much in compensation either, with Milwaukee getting a second-round pick in return.
The one concern here is Dudley's health. Patella issues hampered his 2013-14 season with the L.A. Clippers, who dumped his salary on the Bucks last summer. Dudley credited the Milwaukee athletic training staff with helping get him healthy, though he still missed some time in March and April. The Wizards will have to keep Dudley on the court to maximize his value.
Milwaukee Bucks: C-

I'm surprised the Bucks would move on so quickly from Dudley, whose leadership and bench play were major factors in their surprising playoff berth last season. Dudley had indicated he was interested in signing an extension to stay in Milwaukee. Apparently the front office didn't share that sentiment.
The Bucks do have a crowded roster after dealing for Greivis Vasquez on draft night and signing Greg Monroe as an unrestricted free agent earlier Friday. I'm guessing Milwaukee thinks it can replace Dudley from within. Monroe's arrival may push John Henson to backup power forward, and the Bucks have plenty of options on the wings. In addition to No. 2 pick Jabari Parker, Milwaukee also has Damien Inglis (a combo forward taken early in the second round) coming back from injury.
Still, it's hard to see any of these players being as effective for the Bucks off the bench as Dudley. Among players who saw at least 1,000 minutes of action in Milwaukee last season, Dudley's plus-3.7 net rating per 100 possessions was the team's third-best according to NBA.com/Stats. The Bucks were slightly outscored with Dudley on the bench. So unless a deal offered more value in return, I would have kept Dudley around through the conclusion of his contract.