ARM JTAG is semi-standard and has many open hardware and software implementations.
But, as on many comparable devices you have a lower cost option too:
The Stellaris ® serial flash loader is a preprogrammed flash-resident
utility used to download code to the flash memory of a device without
the use of a debug interface. The serial flash loader uses a simple
packet interface to provide synchronous communication with the device.
The flash loader runs off the crystal and does not enable the PLL, so
its speed is determined by the crystal used. The two serial interfaces
that can be used are the UART0 and SSI0 interfaces. For simplicity,
both the data format and communication protocol are identical for both
In other words, all you really need is a configurable baud rate serial port, and some software which implements the data transfer described in the following portion of the data sheet (which is currently at http://www.ti.com/lit/ds/symlink/lm3s1968.pdf) It would be worth a web search to see if there's is already an open source implementation of a compatible loader utility out there.
For your toolchain problem, code sourcery's GCC download package should probably work, you will just need a linker script suitable for the memory layout of the part, which you can probably find with a web search for projects targeting that part done by other open tools enthusiasts.