"It would be a good chance to study an asteroid while we/they're at it. The problem may be in catching the darn thing for the ride."

One could use gravity assists- one can use, Mercury, Venus, Earth, Mars and even Jupiter.
Most common might be to bounce off Venus, Earth, Venus, Earth as Galileo did:
"The gravity assists from Venus and Earth (two times) that Galileo received on its way to Jupiter caused 11.1 km/sec of velocity change. It would have taken an extra 10,900 kilograms of propellant to get that same boost--about twelve times more than what was on board at launch. "
http://www2.jpl.nasa.gov/galileo/fact/index.html
Or Venus, Venus, Earth as Cassini did:
"Not even the powerful Titan-4B/Centaur can reach Saturn directly with this payload, but it can provide sufficient energy for a direct trajectory to Venus. Here, the great velocity gains from a gravity assist must be used to reach Saturn 6.7 years after launch, by flying by Venus twice and Earth and Jupiter once each - the so-called 'VVEJGA trajectory' (Venus-Venus-Earth-Jupiter Gravity Assist)
...
For the VVEJGA route, the Sun-relative speed gains for each of the four swingbys are roughly 6, 7, 6 and 2 km/s, respectively. "
http://www.esa.int/esapub/bulletin/bullet92/b92kohlh.htm

In addition one might use ion propulsion- which though has low thrust, has greater fuel efficency- plus you going to have to some way of generating energy- solar or nuke, and this source of energy could used for ion propulsion.