Ok, I've just tried the lowpower library with both Mega2560 and UNO boards, and it works fine with both boards. Tang, I really don't think the Arduino board is the problem, you likely have something not hooked correctly. That's my best guess.
You need 7 wires from the Arduino to the RFM12 module, and it's easy to get something crossed. You also need voltage-dividers on at least 3 pins, SCK,MISO,SS. I would recheck the connections at the RFM module, as those pins are easier to mistake than the Arduino pins.
The library is correct for the pin assignments, although it's written in a confusing manner. Eg, in the following for 2560, the 1st line means Arduino D2, the next three are Port.pins on the chip, and the last four are Arduino pins again, D50..D53. The code forces use of INT0, which is the D2 pin.
#if defined(__AVR_ATmega2560__) || defined(__AVR_ATmega1280__)
#define RFM_IRQ 2
#define SS_DDR DDRB
#define SS_PORT PORTB
#define SS_BIT 0
#define SPI_SS 53 // PB0, pin 19
#define SPI_MOSI 51 // PB2, pin 21
#define SPI_MISO 50 // PB3, pin 22
#define SPI_SCK 52 // PB1, pin 20
#elif defined(__AVR_ATmega644P__) || defined(__AVR_ATmega1284P__)
#define RFM_IRQ 10
#define SS_DDR DDRB
#define SS_PORT PORTB
#define SS_BIT 4
#define SPI_SS 4
#define SPI_MOSI 5
#define SPI_MISO 6
#define SPI_SCK 7
#elif defined(__AVR_ATtiny84__) || defined(__AVR_ATtiny44__)
#define RFM_IRQ 2
#define SS_DDR DDRB
#define SS_PORT PORTB
#define SS_BIT 1
#define SPI_SS 1 // PB1, pin 3
#define SPI_MISO 4 // PA6, pin 7
#define SPI_MOSI 5 // PA5, pin 8
#define SPI_SCK 6 // PA4, pin 9
#elif defined(__AVR_ATmega32U4__) //Arduino Leonardo, MoteinoLeo
#define RFM_IRQ 0 // PD0, INT0, Digital3
#define SS_DDR DDRB
#define SS_PORT PORTB
//OLD from Jeelib: #define SS_BIT 6 // Dig10, PB6
#define SS_BIT 0 // Dig17, PB0
#define SPI_SS 17 // PB0, pin 8, Digital17
#define SPI_MISO 14 // PB3, pin 11, Digital14
#define SPI_MOSI 16 // PB2, pin 10, Digital16
#define SPI_SCK 15 // PB1, pin 9, Digital15
#else
// ATmega168, ATmega328, etc.
#define RFM_IRQ 2
#define SS_DDR DDRB
#define SS_PORT PORTB
#define SS_BIT 2 // for PORTB: 2 = d.10, 1 = d.9, 0 = d.8
#define SPI_SS 10 // PB2, pin 16
#define SPI_MOSI 11 // PB3, pin 17
#define SPI_MISO 12 // PB4, pin 18
#define SPI_SCK 13 // PB5, pin 19
#endif
Also, if you look at the 328 assignment at the end, the SPI pins are pretty clear, D10..D13. It's just written in a confusing manner. For better clarity, it would say something more like
#define SPI_SS 10 // Arduino D10, PB2
#define SPI_MOSI 11 // Arduino D11, PB3