Smooth tires on wet road?
I wanted to disentangle a few concepts here as most discussions around tire tread and wet conditions have typically been in regards to hydroplane/aquaplane, which other answers have correctly indicated is not a general problem for bicycles.

While tire tread is not needed to prevent hydroplaning, tire tread can still play a role in traction, explicitly helping the tire to interlock with road imperfections.  In low traction scenarios (such as wet conditions) getting every bit of traction available can be important. 

In general bicycle road tire traction will be related to the following factors: 


Tire compound and environmental conditions (e.g., temperature, wet/dry/snow)
Tire construction (e.g., tire suppleness) and tire pressure
Tire tread pattern


Tire Compound

This is probably the biggest predictor of whether or not you will have traction in the rain.  The quality and tackiness of the compound can vary greatly by price and manufacturer.  If you have a poor quality tire with a lousy compound it will likely be like riding on ice when it gets wet (regardless of whether or not it has tread). The ability for a tire compound to "hook up" will also depend on temperature.  In cooler temperature tire compound will harden up and provide less traction (wet or dry).

From John McKone (Vittoria)


  Compounds come into play and have the most influence on traction (wet and dry conditions); recently developed compounds such as Vittoria’s ISOgrip provide a much broader, stable compound in lower temperatures, this also a key component to a tire’s overall grip in all conditions.
  
  Technical FAQ: Tire grip in wet conditions


Tire Construction  &amp; Tire Pressure

Traction is gained by the tire interacting with the road surface.  If you have a very stiff tire (i.e., not supple) or pump it to a very high pressure the tire rubber will have a hard time interacting with the imperfections in the road surface providing less traction.  In the wet traction is already reduced so this effect can become more important than in the dry where roads provide a lot of traction even under a bad setup.

From Alex Brauns President, Challenge Handmade Tires


  It is critical to fine-tune every component of the tire before riding aggressively in wet conditions to maximize the surface area gripping the road surface while squeezing water out from under the tires wherever possible. This is why all top level road teams and even experienced pro triathletes will ride tubular tires with soft, supple casing materials, natural rubber tread compounds, and supple latex inner tubes, sized to the rider weight and road condition and adjusted to a minimum pressure to keep the rims from bottoming out on the bumps.
  
  Technical FAQ: Tire grip in wet conditions


Tire Tread Pattern

The debate around tire tread vs slick tires in wet conditions has generally focused on aquaplaning/hydroplaning, however on the road a very light amount of tread (e.g., herringbone pattern on the old road tires) can be useful for interlocking. 

Wolf Vormwalde who was the main engineer for Continental tires before switching to Specialized has this to say:


  Common perception is that tread has no influence in road bicycle tires. But this is not entirely true. The question has to be how much tread do you need?
  
  Does tread have an influence on traction in road bicycle tires? Yes. Tread does have an influence on traction in road tires. Tread, if it is not so high that it lifts the tread ground off the road (&lt;0.2mm), helps sink tread rubber into rough road surfaces and penetrate lubricants (water) or dirt. Peaks in the tread increase press into surface crevices and add contact points between the road and the tire, and thus increase friction. Just a roughened or scuffed tread surface passes as a tread already. It does not need to be designed shapes.
  
  Technical FAQ: Tire grip in wet conditions


And from From John McKone (Vittoria)


  Tread patterns may decrease a contact patch by creating deformation of the tire due to road pressure under load from the rider and the rubber/casing bulges and deforms to fill the tread voids, but a well thought-out tread design increases its contact area when turning, accelerating, and braking (increased load due to force will flex the diamonds and grooves) and will also provide micro interlinking in between tread pattern edges and road surface grooves. 
  
  Technical FAQ: Tire grip in wet conditions


Note we are talking about a incredibly light tread pattern (scuffing a tire is enough).  We are not talking about deep channel wells or off-road knobbies.  Also this is likely a secondary effect compared to tire compound which most manufactures have been chasing now for the last few decades.

For another good discussion on light tread pattern and interlocking see Jan Heine's Optimizing Tire Tread article.