The MIT High-Low Tech tutorial on using an Arduino board as an ATTiny programmer has a diagram for connecting an Arduino to program an ATTiny45, but not for programming an ATTiny44. So I cross-referenced the corresponding SCK, MISO, and MOSI pins with the package pins, and came up with this:

That is, connect Arduino pins 10, 11, 12, and 13 to the ATTiny44 package pins 4, 7, 8, and 9 as shown. Ground goes to ground, and the Vcc pin on the Tiny connects to the 5-volt pin on the Arduino power header. (And of course, when I say ‘Arduino pin’ I really mean ‘Arduino header hole,’ but you knew that!)

I tested this with an ATTiny44A-PU and got an LED to blink, so I think this is good. I haven’t tested an ATTiny84, but assume it’s the same thing. Let me know if there are problems.