This is more of a supplemental answer since this accepted anser answer did not touch on the characteristics/attributes of other positions outside of the QB position.
It's more advantageous for linemen in this offense to be quicker, especially in lateral movement, even at the detrement of being better pass blockers or power blockers due to less of an emphasis on traditional dropback pass plays and downhill running plays.
Many spread running plays are all predicated on a stretch zone blocking scheme where the lineman all attempt to block laterally and gain outside leverage on the defender that is threatening the gap to the side they are blocking. The lineman must be athletic enough to be a threat to do this, otherwise there is very little reason for the defense to flow as hard to the side the lineman are blocking to. It's the threat of the lineman getting outside leverage that even allows the QB to option off of a backside defender when choosing what to do.
TEs and FBs
There is more emphasis on both being a good blocker and favoring more quickness... in their case it's so they are able to perform blocks such as arc blocks correctly (see http://www.coachwyatt.com/veerexplained.html).
Athletic and durable enough to be a ball carrier but must still be a threat to be a weapon throwing the ball. Must be able to quickly make reads of defenders in the box during running plays.
Nothing extra noteable.
Required to be better blockers than typical and also better at running routes against man coverage, as opposed to being great at running option routes or finding holes in zones, since the majority of coverages seen will be man coverage with a single high safety. Speed is more of a premium here.
Also, as a bit of a soapbox rant, the term "spread" is really a media invented term used to describe and communicate about a style of offense where the primary goal of the base offensive plays is creating options between a QB/RB and a single defender "in the box" in order for a QB to make a single read and determine, in the middle of the play, what he should do with the ball. This specific concept is really no different than the option offenses that have been run throughout the history of football.
The only thing that makes it interesting now is that the initial alignments in formations resemble what a passing oriented pro style passing offenses look like and that the plays are designed such that interior blockers are engaged in a lateral leverage style of (zone) blocking as opposed to straight ahead power blocking or over reliance on angle and trap blocking.
The intention being to get the defense moving in one lateral direction away from where the QB can be potentially isolated against a defender responsible for containment in the opposite direction of where the defense is moving. The Broncos of the late 90s, Falcons of the early 00s, and the present day Texans, utilized these concepts too and with great success, except they do it under a center and with pre-snap/predetermined reads with stretch zone running plays and QB bootleg action.
If you really want to see a fun evolution of this, watch these videos at http://brophyfootball.blogspot.com/2011/08/alex-gibbs-stretchgun-run-developments.html where Alex Gibbs, the run game guru from the Broncos and Falcons during the 90s and 00s, consults with the University of Florida staff on improving their running game.