Radial lacing is the spokes going straight from hub to rim without crossing another spoke. Pretty common on the front wheels of road bikes (for a slight weight savings); can't remember ever seeing a mtb wheel laced that way.
From Sheldon Brown
"Radial Spoking
Radial spoked (cross 0) wheels have the spokes going straight out from the hub. This pattern is suitable only for front wheels that don't use hub brakes. They are very cool-looking, and are often a good choice for the ultimate in performance, because they are slightly lighter and, in theory, may have a very slight ærodynamic edge.
There are two things to watch out for with radial wheels. Because the nipples point straight inward from the rim, they can turn more easily in most rims than when they are bent to a slight angle by a semi-tangent spoke pattern. This ease of turning increases the risk of their unscrewing themselves on the road. To prevent this, nipples on radial wheels should not be lubricated, and it is a good idea to use a spoke adhesive such as Wheelsmith Spoke Prep or one of the milder flavors of Loctite ® on them. (Or, if a rim does not have recessed spoke holes, the rim tape and air pressure in the inner tube will keep the spokes from turning -- at least with a high-pressure tire).
The other potential problem with radial wheels is that the spokes, trailing straight outward on the hub flange, can possibly rip the outer edge of the flange right off along the line of the spoke holes. This is most likely to happen with small-flange, 36 hole hubs, because there is less metal between the spoke holes. If a used hub is re-laced radially, the notches left by the old spokes can act as stress risers, further weakening the flange.
Many hub manufacturers specifically recommend against radial spoking for this reason, and will not honor warranties on hubs that have been spoked radially.
Some folks will say that no bicycle wheels should be radially spoked for this reason, so do this at your own risk. In my experience, it's generally OK with good-quality hubs that have forged shells.
[Note from John Allen: heed this! Back in 1980, unaware of these issues, I built a front wheel for my tandem with 40 radial spokes on a medium-flange, "boutique" hub with machined flanges. After a few months of use, the wheel mysteriously went slightly out of true, and I trued it. I might better have taken the loss of true as a red-flag warning. One day, I just happened to be sitting on the floor next to my tandem, glanced at the front hub and noticed a crack extending along the line between several spoke holes. I had been foolish, and I was very lucky. I rebuilt the wheel on a large-flange hub, using the same spokes in a semi-tangent pattern, and I have ridden that wheel for 30 years since without any problem. If you ever notice a radially-spoked front wheel -- or, actually, any front wheel -- mysteriously going slightly out of true, stop, get off the bicycle and call for a ride home or to a bike shop.]
If you want to take your chances and try a radial-spoked front wheel, I would advise avoiding using thick spokes and very high tension. Since front wheels are generally pretty trouble-free compared with rears, you don't need super-high tension on a front wheel that uses a reasonable number of spokes.
[Another note from John Allen: I don't recommend leaving spokes slacker than their optimum. If you use the thin spokes, you can raise them to their optimum tension without their putting undue stress on the hub.]
Bicycle folklore has it that radial-spoked wheels give a "harsh" ride, because the shorter spokes are less "stretchy" than the longer spokes used in semi-tangent wheels. This is hooey!
Drive wheels and wheels with hub brakes should never be radially spoked. Due to the near-perpendicular angle of the spoke to the hub's tangent, any torque applied at the hub of a radial-spoked wheel will result in a very great increase in spoke tension, almost certainly causing hub or spoke failure.
[Yet another note from John Allen: I have seen such a wheel. To protect the guilty, I will not say who built it. No, it wasn't Sheldon. A friend and I inspected the bike. He held the front brake and pushed down on a pedal with his foot. The spokes of the rear wheel changed angle noticeably, pinging as they rotated in the spoke holes of the hub, and ringing with rising musical pitch like an electric guitar when the player pulls up on the tremolo bar.]"