I don't have any idea what the three feet of contact is about, all eight feet of the rod has to be in the earth, and yes you can lay the rod in a trench and throw dirt on top of it.
250.53 Grounding Electrode System Installation.....
(G) Rod and Pipe Electrodes. The electrode shall be in-
stalled such that at least 2.44 m (8 ft) of length is in contact
with the soil. It shall be driven to a depth of not less than
2.44 m (8 ft) except that, where rock bottom is encoun-
tered, the electrode shall be driven at an oblique angle not
to exceed 45 degrees from the vertical or, where rock bot-
tom is encountered at an angle up to 45 degrees, the elec-
trode shall be permitted to be buried in a trench that is at
least 750 mm (30 in.) deep. The upper end of the electrode
shall be flush with or below ground level unless the
aboveground end and the grounding electrode conductor
attachment are protected against physical damage as speci-
fied in 250.10.