ZKC Controller experienced loss of separation event between an Air Carrier and Military aircraft when he/she issued a heading to avoid traffic. The Air Carrier; believing assignment was an altitude instruction; began a descent.

Narrative

Air Carrier X was established direct VLA on the VLA6 STAR when he checked on my frequency level at FL240. Military Y; a flight of 3 military aircraft; was already on my frequency established in a block altitude FL220 to 230. The military aircraft were approximately 3 miles East of the Air Carrier X. The aircraft were going to eventually cross out and diverge and I was then going to get lower for Air Carrier X. I issued the traffic to both aircraft and they both reported traffic in sight. Air Carrier X then needed to be turned to avoid traffic in the sector above my sector at FL240. So after getting control from the sector above me I issued direct PETTI intersection; which turned Air Carrier X to a more Southwesterly heading. After the conflict was resolved in the high sector I then realized that Military Y and Air Carrier X were not going to diverge quickly enough to get Air Carrier X down to cross PETTI at 110; so I issued to Air Carrier X 'for your descent fly heading one eight zero.' Air Carrier X replied 'one eight zero'. I then scanned the sector to take care of some other tasks that needed to be completed. Approximately 30 seconds later Air Carrier X questioned me about the descent to FL180!? I told him to maintain FL 240. He then said; 'Oh yeah heading one eight zero.' By this time he had already started his descent and lateral separation was possibly compromised. I was then alerted that the 'snitch' went off and was relieved from position. Recommendation; I think it is completely unacceptable that we as controllers are held accountable for everything that comes out of our mouth; and pilots can fly around like cowboys saying what ever they want. This was clearly a pilot problem; first not listening; and second with the read back being completely unclear. When a heading is issued pilots should have to use a term such as 'turn' or 'degrees.' When an altitude is issued in Class A; pilots should be required to preface it with 'flight level.' When not in Class A they should have to follow the altitude read back with something like 'feet'. We are expected to hear read backs; well the read backs should be clear.