You know, I wonder. Could the R4 handle microSDHC cards with the right software (as in, a new firmware update)? Depends on whether the microSD driver is in hardware or software. I'm guessing hardware, but you never know. Let me look up some IC numbers.
Unfortunately, the ATF01 chip that is the heart of the R4DS is not documented at all, so we'll never know for sure. BUT, I can tell from the way the R4DS behaves that this ATF01 has a bootloader in some kind of nonvolatile memory (burned ROM? EEPROM? Flash memory? who knows) that then initializes the SD card and boots _DS_MENU.DAT, the actual firmware. (PC technicians: sound familiar?) This configuration has a noted advantage, that is one of the R4DS's selling points -- the actual firmware is entirely contained within _DS_MENU.DAT on the microSD card; replacing _DS_MENU.DAT has the net effect of completely replacing the R4DS's firmware, and if anything goes wrong, you can always rewrite the file using an external card reader, put your card in your R4DS, and boot it right up. Thus, firmware upgrading is completely risk-free. However, this has the limitation that if the bootloader can't read the card, then it can't boot the real firmware. And since the bootloader doesn't support SDHC, you need to have a standard microSD card in the R4DS when it boots for much of anything to happen.
There's several possibilities. The most obvious is replacing the bootloader, by flashing the memory it's stored in. However, this has a big catch: if the flash goes wrong, and the DS is reset before you get a chance to repair the damage, then you just bricked your R4DS. The aforementioned selling point thus goes straight out the window. A safer approach would be to simply flash the updated bootloader onto new R4DS's as they leave the factory, and optionally run a recall on existing units. The latter is the most likely approach Team R4 will take, if they intend to add SDHC support to R4DS at all.
And then there's the bootstrap approach. In most other platforms that use the BIOS->bootloader->OS bootstrap sequence, once the bootloader has handed over to the OS (in this case, the R4's actual firmware, aka _DS_MENU.DAT), the OS is talking directly to the hardware, and the bootloader completely leaves RAM. At this point, if the OS supports it, you could remove the standard microSD card you booted the R4 with, put in the microSDHC card, and let the OS's SDHC driver boot the card and read it. This has the (probably major) inconvenience of having to constantly switch cards back and forth.
Obviously, none of this can happen unless and until Team R4 takes the initiative and follows one of these paths. All we can do is wait.
It works with disks because the disks are designed to be backwards compatible. SDHC cards are *NOT* backwards compatible. It won't work; you can only read SDHC cards in SDHC readers, regardless of how the card is partitioned.