Short version: in the current Standard environment, play Sphere of the Suns if you want to play 4-drops early, play Koth of the Hammer if you want to play 6-drops early or get a lot of mana for a big X spell; make sure you don't short yourself on lands; if you want to experiment with a rogue deck, try Geosurge.
In the modern color pie, red is the color for fast-mana effects (called "rituals" after the old Dark Ritual). Unfortunately, the two ritual effects available in Standard, Geosurge and Infernal Plunge, don't quite hit the same sweet spot as Seething Song used to. Geosurge is still good enough to drop a big finisher (Inferno Titan, Wurmcoil Engine, maybe Balefire Dragon if you want to get cute) on turn 4, but that requires having both the enabler and the big creature in your starting 10 -- and remember that contemporary mono-red doesn't really have filtering or card draw. Most people probably figure it's safer to just play a strong four-drop.
I know some folks have experimented with B/R Heartless Summoning to play Inferno Titans quickly (turn 3 Titan using Priest of Urabrask, or turn 3 Urabrask into turn 4 hasty Titan). Those decks are rather inconsistent and vulnerable to disruption, though.
Honestly if you need consistent early-turn mana acceleration in a color other than green, I'd just play Sphere of the Suns. Like Myr, it's still an easy target for Ancient Grudge, but at least with Sphere half your mana base won't be blown up by a Day of Judgment.
Solemn Simulacrum is a great card for getting you from 4 mana to 6. As a bonus, you get to draw a card when it dies. That's nearly always a 2-for-1, and often a 3-for-1 if you can trade creatures.
The best big-mana card in red is Koth of the Hammer. His -2 ability gives you a lot of mana easily (if you have Koth and 5 lands on turn 5, for instance, you can play Inferno Titan and have mana open to stop Mana Leak, or hit your opponent with a 9-point Devil's Play). Koth's -5 ability is excellent at breaking through board stalls.
Also, with all these tricks, it's important to remember that ramp doesn't replace land drops. Most red decks running Koth as their big threat will want to consistently hit their 4th land drop on turn 4, and that means running 24 lands rather than the 20-22 you can sometimes get away with in aggro decks.