HDMI accepts DVI signals; all you need is a passive adapter, to connect a DVI-D output to an HDMI input.
Late edit: I was wrong - they're both serial.
That said last year I needed HDMI output used a tang nano 9k (£10) and GOWINs free IDE and IP cores were a breeze to use.
The DVI core basically takes what I remember being a VGA signal.
Their HDMI core IIRC just took a framebuffer.
So my complaining is probably in the vein of old many yells at cloud.
The analog pins on the DVI connector were used on many video cards for output, but they were mostly there so that you could use a DVI-to-VGA adapter, not because there were many devices which could use it directly from that pinout.