Thanks a lot for your responses.
Since you haven't provided any information at all about your standalone system a few questions come to mind.
What are you using for a power supply on your standalone system?
Sorry that I didn't make it clear. With standalone I mean a bear Atmega 328 on a Breadboard as described on the Wiki. I'm using it with the external 16 MHz-clock which seemed irrelevant to mention for me. Sorry about that.
To power the whole thing I'm getting power directly out of the USBasp I'm using to program the device.
Have you adequately bypassed all of the power pins on your standalone system?
I'm sorry, I'm not sure what you mean by that. If you'd wanted to ask if I connected all GND-Pins on the Atmega to GND and all VCC-Pins to VCC then I'd say: yes, I did.
You tried it with a Micro (32U4), and ported it to a 328.
So you did change the code to work correctly on the respective chip.
Did you also take the pins that offer PWM in consideration, which is likely to be different ?
I'm not sure what changes you mean by that else than changing the pin. I used Pin 9, which is a PWM on both chips.
Also, did you check what current is drawn by the display and take measures for the current to not exceed the maximum allowed current per pin and / or total current for the device ?
You might need to use a transistor to handle the needed current, and the typical setup of that requires the dimmer output of the chip to be inverted (you can easily do that by software).
I already tried a transistor with a 2.2k-resistor on the base (and directly powering the Backlight, not inverted yet). But that had the same effect.
While writing this I think I narrowed down the issue. When powering everything via a Genuino Micro (5V- and GND-Pin via the same USB-Port) or a Duemilanove Board (without Atmega) everything works fine. Only while powering it over the USBasp (also when everything else -- i.e. MOSI, MISO, SCK, RST -- is disconnected) the flickering happens. But as soon as I connect my contraption with the unpowered Arduino Duemlinanove, everything plays nicely (i.e. power from the ISP, but GND and VCC lines connected to a Duemlianove board without USB-connection and processor). What is there on the board that helps in this case? Unfortunately my understanding of the USBasp isn't as deep to understand this.
For reference: I bought my ISP from a no-name-trader. After research I found that it must be this programmer. It has a maximum current of 500mA which shouldn't be taken by the LCD as I can drive it with a normal digital pin on the Genuino Micro (max. of 20mA) or the Atmega (without using PWM).
I'm really thankful for any further ideas on what's the issue.