"Fly runway heading" is a vector, and many towers are not permitted to provide vectors except to IFR aircraft when the overlying approach control authorizes it. This is why you'll hear a lot of towers (especially in DFW) saying things like "Proceed westbound" instead of "Fly heading 270." It is important to note that in the context of the OP's post, a vector would not be appropriate, as it would not account for wind aloft which could push the aircraft into the path of arriving traffic.
From the AIM (which is more relevant to us than the 7110.65):
"The local controller may provide pilots flying VFR with generalized instructions which will facilitate operations; e.g., “PROCEED SOUTHWESTBOUND, ENTER A RIGHT DOWNWIND RUNWAY THREE ZERO,” or provide a suggested heading to establish radar identification or as an advisory aid to navigation; e.g., “SUGGESTED HEADING TWO TWO ZERO, FOR RADAR IDENTIFICATION.” In both cases, the instructions are advisory aids to the pilot flying VFR and are not radar vectors. NOTE− Pilots have complete discretion regarding acceptance of the suggested headings or directions and have sole responsibility for seeing and avoiding other aircraft."
From the 7110.65:
"It is important that the pilot be aware of the fact that the directions or headings being provided are suggestions or are advisory in nature. This is to keep the pilot from being inadvertently misled into assuming that radar vectors (and other associated radar services) are being provided when, in fact, they are not."
If we can handle "Make straight-in Runway 34," which is an approved instruction, we should be able to handle "Fly straight out. Runway 34, cleared for takeoff." It simply means to continue on the upwind leg. It does not mean "fly runway heading."