What exactly defines modern? Are we talking about weapons we already make, or weapons we currently have the capability to make? I'm asking because 'tis currently possible to create things such as railguns and anti-aircraft lasers. Not to mention, what counts as a weapon? Things intended for agriculture or construction can be used as effective weapons (ie. sandblaster). You also have to consider the variety of dragon being dealt with. Obviously, some are more difficult to kill. Last but not least, if dragons exist in this hypothetical situation, does magic? When it comes to killing dragons, I'd likely say an 88mm tank shell or a good shower of 7 to 8 shots from a MK 19 should do the trick for most dragons. Anything beefier, and 'tis like medicine; the higher the weight, the more the dosage.

Snipe it's eyes with some high caliber rifle, then while it is blinded, assuming it is on the ground once it gets shot, run at it and toss a couple grenades in its mouth. Problem solved.

Do it Gears of War style and: 

A) Call down the Hammer of Dawn and melt it's armored flesh.

B) Unload a clip inot that f#%$# with the Lancer and jump onto it's back, tearing into it's neck with the chainsaw bayonet until the nerves and arteries are severed. Blood everywhere, and Marcus Fenix has terminated a dragon like a badass.

C) Use a mounted MG turret to shoot it out of the sky, and then finish it off on the ground with a charged sniper round between the eyes. Hooray for OP snipers.

Better yet, do it with a hop and a skip and a dropkick to the snout, climb onto it's back, ignite your lightsaber, electrify it with an Angel Sword modification, slide down it's back and part it's back in two. Then for an added effect place five charges in it's now halved back, walk away, and detonate. Bloody dragon chunks fly everywhere.

As you can see, I've thought a lot about slaying dragons with all sorts of technology... Hey... I have an excellent idea for a thread...