The TLC5940 http://www.ti.com/lit/ds/symlink/tlc5940.pdf can provide constant current
upto 120mA per output (set by single resistor for all outputs). As it has 16 outputs you can parallel
them in threes to give 5 outputs of upto 360mA, and you can program PWM values into it for dimming
(and store them on on-chip EEPROM!!). Only one device for 5 LEDs then.
You'd need to be very careful not to exceed the power dissipation though - perhaps adding enough series resistance to keep
the outputs around 1.0 to 1.5V (below 1.0 the constant-current behaviour deteriorates, the higher the voltage
the more heat).