Peter R. wrote:
"Departing rwy 24, climb runway heading to 1,200 feet (about 500 ft AGL),
then climbing right turn direct to DKK VOR (VOR is on the field) before
proceeding on course."

Here is a small JPG of the relevant sectional chart showing the airport:

http://thericcs.net/aviation/misc/DKK.jpg

My question is this: I am departing runway 24 with a desired on course
heading of 080. Thus, I depart and climb to 1,200 msl, then climbing turn
to the right to go direct to the on-field VOR before proceeding at 090 on
course.

Climbing runway heading at about 800 fpm in a Bonanza, I reach 1,200 feet
MSL in about 38 seconds. 38 seconds of traveling at a ground speed of
about 95 kts does not place me far enough away from the airport to be able
to perform a standard rate turn to the right to go direct to the VOR.

I see your dilemma, but I think you're worried about stuff that
doesn't need worrying about. By the time you started your right turn
at 1200, you were already above anything along your departure path
(and still climbing, presumably). Make the initial right turn to 090
and off you go.

It only gets interesting when heading south and you need to make sure
you clear the 2849 tower (conveniently located smack in the middle of
the airway). In that case, what I would do is set 166 on the OBS and
if I wasn't at the 2300 required by the DP when the flag flipped from
TO to FROM, I'd do one lap in a racetrack pattern (even easier with a
moving map GPS).