Across:
   1.  Big hitters were beheaded and they fell (7)
   5.  Unhappy is initially their style - perhaps they like to hurt (7)
   9,10. They could make things faster by reforming illogical parts (9 5)
  11,12. Mach I race is fun - crash and you pay the ultimate price (5 9)
  13. About Guevara meeting with the right - Che is exotic! (9)
  16.  Fruit pressed as a necessity (4)
  18.  Nations and civilizations alike end thus - they crystallize into salt (4)
  19.  Asked for people's opinions about abstract art, and went around checking up (9)
  22,24.  Do costlier race preparations give a false show of concern? (9 5)
  26,27.  Fingers rub font furiously in 12 (5 9)
  28.  Maidens go wild for guest musician (7)
  29.  Little lumps of lode sun has broken down (7)
Down:
   1.  Leap there, abandoning tails for sturdier garb (7)
   2.  Send me and my diary back - figurine said to come alive (5)
   3.  Construct motor with elliptic rotating heads (8)
   4.  Hears, sees and locates (5)
   5.  Toughest 4: tract without a development (9)
   6.  Tied up around church - it's a trick (6)
   7.  Music to suit April Fools (9)
   8.  Cuss pet ("Bad Puss!" etc.) as who you think might be responsible (7)
  14.  Smooth Count rode about (9)
  15.  Cairns' crop is processed with no help from the South - surely a sign! (9)
  17.  Pete trod carefully, and attended to various small tasks (8)
  18.  Nightmare fellow incubates Torus without a test or ... (7)
  20.  ... stops using design, or Ant's Adventists' text, discarding ignorant sad vent (7)
  21.  Bass part heard to be double time during big burst of noise (6)
  23.  Big Wheel finally turns out something for the little guy (5)
  25.  Try to get rid of pain in guilt every second (5)